A car travels a distance of \(x\) with speed \(v_1\) and then same distance \(x\) with speed \(v_2\) in the same direction. The average speed of the car is

Net gravitational force at the centre of a square is found to be \( F_{1} \) when four particles having mass M, 2M, 3M and 4M are placed at the four corners of the square as shown in figure and it is \( F_{2} \) when the positions of 3M and 4M are interchanged. The ratio \( \frac{F_{1}}{F_{2}} \) is \( \frac{\alpha}{\sqrt{5}} \). The value of \( \alpha \) is __________ .

A solenoid of \(1000\) turns per metre has a core with relative permeability \(500\). Insulated windings of the solenoid carry an electric current of \(5\, A\). The magnetic flux density produced by the solenoid is (permeability of free space \(\left.=4 \pi \times 10^{-7} H / m \right)\)

Given below are two statements: Statement \(I\) : A uniform wire of resistance \(80\,\Omega\) is cut into four equal parts. These parts are now connected in parallel. The equivalent resistance of the combination will be \(5\,\Omega\). Statement \(II :\) Two resistance \(2\,R\) and \(3\,R\) are connected in parallel in a electric circuit. The value of thermal energy developed in \(3\,R\) and \(2\,R\) will be in the ratio \(3:2.\) In the light of the above statements, choose the most appropriate answer from the options given below

Figure shows elliptical path \(abcd\) of a planet around the sun \(S\) such that the area of triangle \(csa\) is \(\frac {1}{4}\) the area of the ellipse. (See figure) With \(db\)  as the semimajor axis, and  \(ca\) as the semiminor axis. If \(t_1\)  is the time taken for planet to go over path \(abc\) and \(t_2\) for path taken over \(cda\) then

Consider a water tank as shown in the figure. It's cross-sectional area is \(0.4\, m ^{2}\). The tank has an opening \(B\) near the bottom whose cross-section area is \(1\, cm ^{2}\). A load of \(24\, kg\) is applied on the water at the top when the height of the water level is \(40\, cm\) above the bottom, the velocity of water coming out the opening \(B\) is \(v\, ms ^{-1}\). The value of \(v\), to the nearest integer, is ......\(m/s\). [Take value of \(g\) to be \(10 \,ms ^{-2}\) ]

A particle of mass \(m\) is moving in a straight line with momentum \(p\). Starting at time \(t = 0\), a force \(F = kt\) acts in the same direction on the moving particle during time interval \(T\) so that its momentum changes from \(p\) to \(3p\). Here \(k\) is a constant. The value of \(T\) is

The figure shows a Young’s double slit experimental setup. It is observed that when a thin transparent sheet of thickness \(t\) and refractive index \(\mu \) is put in front of one of the slits, the central maximum gets shifted by a distance equal to \(n\) fringe widths. If the wavelength of light used is \(\lambda ,\,t\) will be

A body of mass \(2\,kg\) is initially at rest. It starts moving unidirectionally under the influence of a source of constant 10.Power P. Its displacement in \(4\,s\) is \(\frac{1}{3} \alpha^2 \sqrt{ P } m\). The value of \(\alpha\) will be \(.............\)

A parallel plate capacitor is of area \(6\, cm^2\) and a separation \(3\, mm\). The gap is filled with three dielectric materials of equal thickness (see figure) with dielectric constants \(K_1 = 10, K_2 = 12\) and \(K_3 = 14\). The dielectric constant of a material which when fully inserted in above capacitor, gives same capacitance would be

A meter bridge setup is shown in the figure. It is used to determine an unknown resistance \(R\) using a given resistor of \(15\,\Omega\). The galvanometer \((G)\) shows null deflection when tapping key is at \(43\,cm\) mark from end \(A\). If the end correction for end \(A\) is \(2\,cm\). then the determined value of \(R\) will be__________ \(\Omega\)

Electric field in a region is given by \( \vec{E}=Ax\hat{i}+By\hat{j} \), where \( A=10~V/m^{2} \) and \( B=5~V/m^{2} \). If the electric potential at a point (10, 20) is 500 V, then the electric potential at origin is __________ V.

A thin uniform rod of length \(2\,m\). cross sectional area ' \(A\) ' and density ' \(d\) ' is rotated about an axis passing through the centre and perpendicular to its length with angular velocity \(\omega\). If value of \(\omega\) in terms of its rotational kinetic energy \(E\) is \(\sqrt{\frac{\alpha E}{ Ad }}\) then the value of \(\alpha\) is \(...........\)